Draft version September 4, 2014Preprint typeset using LATEX style emulateapj v. 05/12/14

CALCULATING A STELLAR-SUPERMASSIVE BLACK HOLE MICROLENSING RATE

Isabel LipartitoAstronomy Department, Smith College, Northampton, MA 01063

and

Andrea Ghez, Gunther Witzel, Breann Sitarski, and Samantha ChappellAstronomy Department, University of California, Los Angeles, Los Angeles, CA, 90095

Draft version September 4, 2014

ABSTRACT

The Galactic center contains a supermassive black hole whose gravitational fieldtidally affects nearby objects. One observable effect is relativistic stellar microlens-ing, or the deflection of stellar light from orbiting stars passing in the vicinity ofthe black hole. This leads to the generation of multiple unresolved stellar imagesand an overall increase in brightness of the source. Detection and analysis of mi-crolensing events give information regarding the nature of the black hole and theworkings of general relativity. We aim to calculate an updated rate of observablemicrolensing events for K-Band emitting stars present in the innermost few parsecsof the Galactic center, taking into account current data available from the UCLAGalactic Center Group for stellar densities, velocities, and luminosities. Lensingevent rates are calculated both for contemporary observational limits, specifically,those for Keck Observatory and its current adaptive optics system, as well as thelimits for telescopes of the future (the Thirty Meter Telescope). Furthermore, aneffort is made to take into account the duration of observing runs. This calculationwas originally completed in 1999 by Alexander and Sternberg; rates generated sug-gested detectable lensing events are one event per one thousand years, too rare forus to expect to see on observation timescales of years or even decades. However,with the improved observation parameters of adaptive optics, we can expect todetect one lensing event per 5 years (via Keck AO) for unresolved lensing in thecase where observations are spaced by several months or years apart. The newgeneration of telescopes will have even greater detection potential, as projecteddetection rates are one event every 1-2 years even without continuous observation(time spacing of months or even years).

Subject headings: galactic center, microlensing, lensing, black hole, SMBH

1. INTRODUCTION

Multiple routes exist by which one can gatherinformation indirectly about the environmentof the Galactic center and its workings, gov-erned by gravitational interactions between theGC Supermassive black hole of 4.2 million solarmasses, Do et al. (2013). One of these is thedetection of SMBH microlensing events. TheGalactic center contains a dense population ofS stars orbiting the black hole, whose orbital as-trometry has previously been used to constrainmeasurements for the mass and distance of theblack hole from Earth, Ghez et al. (2008). We

should expect to thus be able to observe lens-ing events which occur when these stars passin the vicinity of the black hole. Lensing is arelativistic phenomenon requiring a lens (here,the SMBH) and a source (the orbiting star).Photons from an orbiting star follow geodesicsin the warped space-time around the massiveblack hole. If the source and lens are linedup directly in our line of sight, we should ob-serve an Einstein ring of multiple images of thesource, all formed as the photons coming fromthe source follow different paths in space-time.Indirect alignment of source and lens relative to

2 Lipartito et al

our line of sight leads to the creation of two ormore lensed images. In this situation, however,the effect we most often shall observe is knownas microlensing. The Einstein Radius gives thetypical distance between gravitationally lensedimages, Alexander & Sternberg (1999) ,

RE =

√4×G×MBH

c2× r ×DBH

r +DBH

Where r is the lens− source distance

Where DBH is the distance to the SMBH (1)

but the optical resolving power of the telescopeused to observe the Galactic center determineswhether or not we are able to resolve the multi-ple images. The Keck telescope has an angularresolution threshold of 0.06 arcseconds, trans-lating to an lens-source distance of about 6 pcusing a distance to the BH of 8.46 kpc. If welimit our obervations of stellar lensing to con-sist of stars within the inner 0.5 pc radius ofthe GC, then all lensing we can contemporarilyobserve is microlensing, where the source andimages cannot be separately resolved, and theobservational effect of lensing is a brighteningof the source during the lensing event, Alexan-der & Sternberg (1999).

In 1999, a microlensing rate for the innerGC was computed by Alexander and Stern-berg using contemporary values for black holemass, distance from Earth, and the stellar pop-ulation in the GC (stellar densities, velocities,and luminosities). An unfortunately small rateof detectable lensing events (limited by tele-scope resolution in the pre-AO era) was com-puted: about 1-3 events per 1000 years, evenin the limit of continuous observation. How-ever, times have changed. In 2014, we havebigger telescopes (e.g. the twin Keck 10 m tele-scopes), adaptive optics, and greater optical re-solving power. We are able to see even faintermagnitudes and we have updated parametersfor the SMBH mass, distance, and propertiesfor orbiting stars in the GC from the UCLAGalactic Center Group. We thus decided toupdate the rate calculations of 1999 taking intoconsideration the new parameters and data onthe Galactic center. Our calculations demon-strated we should be seeing around 1 event per5 years in the inner GC using our current gen-eration of telescopes and 1 event per year in theinner GC with new telescopes such as TMT.

2. CALCULATING A MICROLENSING RATE

2.1. Basic Equation

Integral to the generation of a microlensingrate is correctly understanding which param-eters need to be part of the rate calculationand how to implement them. The microlensingrate depends both upon parameters regardingthe behavior of stars and relativistic lensing inthe GC along with a few parameters describ-ing observational limits. The calculable lensingrate depends, basically, on the velocities of or-biting stars (how often are they going to passnear enough the black hole for lensing), the ro-tational velocity of the galaxy, and the numberdensity of these stars (how many of them areavailable for lensing). Each of these parametersare functions of distance from Galactic Cen-ter. Another factor in this calculation is thebrightness of stars: the expected average K-Band magnitude of stars in the region we areobserving along with another parameter desig-nating the highest K-Band stellar magnitudea telescope can successfully detect. Anothersignificant parameter determining the rate isthe Einstein Radius of a lensing event, whichitself depends on the distance between a lensand source. Now comes the question of howto combine these basic parameters and how todeal with the distance consideration.

By making the assumption that the SMBHis directly at the Galactic center, then the lens-source distance variable in the Einstein Radiusparameter would be simply the line-of-sight dis-tance between the star and the Galactic center.Excluding the luminosity parameter, all param-eters now depend on distance from the Galacticcenter. As we include a larger distance range,more stars and thus potential lensing eventswill be included in our scope of observation.Moreover, these parameters individually do notdepend on each other. Therefore we multiplythese parameters together and integrate overthe distance from the Galactic center to a speci-fied line-of-sight distance, whose magnitude de-pends upon what sort of lensing we want tosee. Specific to each instrument, a certain lens-source distance limit will enable us to only seemicrolensing (call this the ’microlensing limit’);further beyond that point includes lens-sourcedistances and hence lensing events that can beresolved. Below is a basic template for lensing

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 3

rate calculations introduced by Alexander andSternberg. Included is a parameter for the stel-lar velocity field, stellar number density, andEinstein Radius, all dependent on the distancefrom the lens.

Γ(K0) = 2× 〈u0〉 ×∫ r2

r1

RE × v × n∗ dr (2)

2.2. Observational Parameters

The aforementioned parameters, however,only take into consideration the astrophysicalparameters at the Galactic center. We need tobe concerned not only with the total number oflensing events that are occurring over a certaintime period, but how many of those events weare able to detect, as well as the nature of theseevents. We need to know the threshold bright-ness of our telescope (how faint a star can wesee).

〈u0〉 = 10K0−〈K〉−2.5 (3)

Additionally, it is necessary to know the meanimpact parameter, u0 defined as the ratio ofthe average flux from a star divided by the de-tection threshold flux, Alexander & Sternberg(1999). (K0 is the threshold K magnitude cor-responding to the treshold flux, 〈K〉 is the av-erage stellar magnitude corresponding to theaverage stellar flux.)

We need to know the lens-source distancelimit for microlensing as well. This value can beobtained by calculating the lens-source distancethat will generate an Einstein Radius at the op-tical resolution of the telescope in question; theminimum distance for resolved lensing.

Solve for r :

arctan[φ]×DBH =

√4GMBH

c2r ×DBH

r +DBH(4)

2.3. Rotational Velocity and Time Dependence

In adddition to the peculiar motions of starsin the galaxy,

vrotational = 80 + 20 log[r

1 pc] (5)

it is important that we take into accountthe overall rotational velocity of the galaxy.Alexander and Sternberg provides a function

to work with this:

G[r] = e−v2rotational2

2 ×∫ vmax

0

y2×√

1− (y

vmax)2

× e−y22 × I0[y × vrotational2] dr

Where I0 is a 0th order modified Bessel function(6)

(Note: rotational velocity here, vrotational2, isrotational velocity divided by peculiar velocity,to be made unitless for this function)

Ak = 2.6 Extinction Coefficient

(7)

∆ = 5× log[DBH ]− 5 Distance Modulus(8)

us =1

10−.4×(23−〈K〉−∆−Ak(9)

vmax =2× us ×RE

v × T(10)

Another parameter, vmax, goes into this func-tion. This is where the time dependence of ob-servations comes in. In the case of searchingfor lensing events, the yearly lensing rate mustdepend on the period of time between observ-ing runs, as some lensing events will occur inthat time period and will be undetectable tous. The same function that takes care of ro-tational velocity also includes a parameter formaximal event velocity: basically, Einstein ra-dius divided by the time between observations.Events occurring at a velocity at this value orhigher will happen while the telescope is notin use and will not be observed. This func-tion, called G(r) in Alexander’s paper, takes ina unitless rotational velocity and maximum ve-locity and generates a unitless constant to gointo the lensing equation, filtering out eventsunavailable to us and thus accounting for an-other limit of observation.

3. REPRODUCING ALEXANDER ANDSTERNBERG, 1999

Putting this all together, we end up with

2×〈u0〉×∫ r2

r1

RE [r]×v[r]×n∗[r]×G[r] dr (11)

4 Lipartito et al

where v[r] is the 1-D velocity dispersion andn∗[r] is the stellar number density adapted fromAlexander’s paper. RE , G[r], and 〈u0〉 are asdiscussed earlier. Taking into account an opti-cal resolution given for the Keck Era of about120 milliarcseconds, we find that the outer limitfor distance from Galactic center for microlens-ing is 45 pc. Alexander extends the calculationfor total lensing (microlensing and resolved) to300 pc total.

Figure 1 depicts the KLF created and used byAlexander, which predicts an average magni-tude for GC stars of about 20.2. Alexander alsosuggests a threshold magnitude of 17. Usingsuch parameters and values, I worked to recre-ate the 1999 time-dependent lensing rates gen-erated by Alexander and was successful, withinthe same order of magnitude. Most impor-tantly, I was able to recreate the significant re-sult, for a threshold magnitude of 17, a totallensing rate of about 3 events per year.

Figures 2, 3, and 4 show the reproducedmicrolensing (45 pc integration limit) and to-tal lensing (300 pc limit) curves, respectively.These curves are dependent on times betweenobserving runs which range from 0.01 years to10 years. Calculations were done for a 16-19magnitude threshold range, although 17 wasthe accepted threshold at the time.

4. UPDATING THE CALCULATIONS OFALEXANDER AND STERNBERG, 1999

Despite the rather unfortunate 1999 resultof a couple lensing detectable events per mil-lennium, we found it a promising endeavorto recreate the work of Alexander and Stern-berg, taking into account updated parame-ters for black hole properties (mass, distancefrom Earth), and stellar parameters in the GC:stellar velocities, number density distribution,and luminosities, adapted from data and re-sults from the UCLA Galactic Center researchgroup. Most importantly, observational pa-rameters have changed. Since 1999, we havethe advent of adaptive optics for Keck and theprospect of the construction of the Thirty Me-ter Telescope in the future. There should be afar larger number of detectable lensing eventsas we can see even fainter stars. Observationalinstruments considered were Keck and currentAO, Keck and Next Generation AO (NGAO),and the TMT.

TABLE 1SMBH and Stellar Number Density Distribution

Parameters from Do et al. (2013)

Variable Definition Valuerb Break Radius for ρ∗ (pc) 1.51γ Inner Slope of ρ∗ .22δ Sharpness in ρ∗ transition 6.81α Outer Slope in ρ∗ 6.31MBH BH Mass transition (× 106 Solar Masses) 4.62DBH Distance to GC/BH (kpc) 8.46

TABLE 2Average Velocity Measurements from Do et al.

(2013)

Variable Definition Valuevx Central Velocity in X-Direction (km/s) 23.13vy Central Velocity in Y-Direction (km/s) 9.40vz Central Velocity in Z-Direction (km/s) -6.41

4.1. Updated Galactic Center Dynamics

ρ∗[r] = (r

rb)−γ × (1 + (

r

rb)δ)

(γ−α)δ (12)

A new stellar density distribution wasadapted from Do et al. (2013) for stars in theinner GC. I normalized this distribution usingthe estimated total mass of the GC withinthe inner 1 parsec Schodel et al. (2009) toget a working density for stars per cubic parsec.

Along with a normalization constant, thenumber density includes a factor of 0.2, used byAlexander and Sternberg to separate the starsthat emit in the K-Band, from the total num-ber of the stars. K-emitting stars are the onlystars of any use to us as they are the only starswe observe.

n∗[r] = 0.2∗Anormalization constant∗ρ∗[r] (13)

In addition to a new number density, an aver-age three-dimensional stellar velocity was usedfrom Do et al. (2013).

v =√

23.132 + 9.402 +−6.412 (14)

4.2. Updated K-Luminosity Function

In choosing a new luminosity function, onehas to consider which telescope/era to be used.Figure 5 shows an updated KLF for the KeckAO era, taken from W. M. Keck Observatorysprojected NGAO Facility, 2014 (MSI (2014)).

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 5

TABLE 3Summary of lensing rates for different

telescopes and integration limits

Eventsperyear

Keck: Pre-AOK Thresh: 17〈K〉 Mag: 20.2Micro. Lim: 45 pc

Keck: AOK Thresh: 18.5〈K〉 Mag: 17.15Micro. Lim: 7 pc

Keck: NGAOK Thresh: 22〈K〉 Mag: 19.7Micro. Lim: 6.5 pc

TMTK Thresh: 23〈K〉 Mag: 21.35Micro. Lim: 0.72 pc

TMTK Thresh: 24〈K〉 Mag: 21.35Micro. Lim: 0.72 pc

Inner0.5 pc

0.00055 0.036 0.088 0.048 0.12

Micro.Limit

0.0028 0.19 0.45 0.08 0.204

10 pc 0.0028 0.19 0.45 0.25 0.628

50 pc 0.0028 0.19 0.45 0.25 0.628

The blue curve is the KLF for current AO andan average magnitude of 17 has been foundalong with a threshold magnitude of about 18.5(taking into account KLF incompleteness, orwhere

the KLF begins to drop off in the range ofstars we cannot see). The red curve is the KLFfor Next Generation Adaptive Optics, with anaverage magnitude of about 19.7 and thresh-old of 22 mag. Figure 6 depicts a KLF forthe TMT era. This KLF was adapted fromYelda et al. (2013) with an average magnitudeof about 21.35 and threshold of 23-24 mag.

It is important to mention that the ba-sic structure of Alexander’s rate equation wasmaintained, consisting of the integral form, ar-rangement of parameters, and consideration ofobservational time dependence. Individual pa-rameters were revised as has been discussed andplaced back into the skeleton structure of theequation.

5. RESULTS AND DISCUSSION

After replacing the parameters, the samecalculations were run again: for Keck AO,Keck NGAO, and TMT. Care was made tocalculate the individual angular resolutions ofthese telescopes and their corresponding limitswhere microlensing becomes resolved lensing.The angular resolution of Keck AO was con-sidered to be 60 mas, while the same valuefor Keck NGAO and TMT was considered tobe 55 mas and 18 mas, respectively. Thesevalues were calculated using the equationfor the angular resolution of the circularaperture, θ = 1.22 × λ

D where the midpointwavelength for the K-band is 2.2 micrometers(the aperture diameter for Keck is 10 m andthe aperture diameter for TMT

is, of course, 30 m). Table 3 above summarizeslensing rates in the limit of near-continuousobservation for multiple telescopes and dif-ferent values for distance from the Galacticcenter.

Out of curiosity, I recalculated the lensingrate for the pre-AO era (the same magnitudelimits Alexander had used), and I changedthe velocity distribution, black hole parame-ters, and stellar number density. Interestinglyenough, the rate is still only a couple of eventsper thousand years. Moving on to Keck AO andNGAO era, we can expect to see 1-2 events ev-ery 5 years (if we include stars dwelling outsidethe inner few pc beyond the SMBH). Movingon to TMT era, we can practically expect to seeone event per year extending outside the innerfew pc behind the SMBH. It appears that ourupdated knowledge of the GC and occurringlensing events isn’t what is bringing up the rate;it is the improved observational limits and in-struments. With TMT, we can even expect tosee multiple resolved events, as the microlens-ing limit lies in the inner 1 pc but the lensingrate continues to increase out to the inner 10pc. For all instruments outside of the Keckpre-AO limit, if we extend at least 10 pc be-hind the SMBH in our observations, we shouldexpect a lensing rate two orders of magnitudehigher than that calculated by Alexander andSternberg. These results are very promising:it is likely we will be able to observe multi-ple events over the next few decades, especiallywith the advent of TMT. The rate was calcu-lated for all instruments considering only theinner 0.5 pc beyond the SMBH as that is theradial range of the data of the UCLA GalacticCenter Group. Unfortunately, for Keck AO,the rate is only about 1 event per 30 years. It

6 Lipartito et al

is not likely we will see more than one event (ifthat) out of a dataset only spanning 20 years.

Figure 7 depicts a few plots which illustratethe time-dependent nature of the lensing rates.The revised calculations for pre-AO Keck havea strong dependence on the time between ob-serving runs, whereas the AO and NGAO cal-culated rates remain rather constant with theincrease in time. It appears that the improvedobserving parameters allows for the detectionof a higher proportion of lensing events thatare longer in duration, which would weakenthe time dependent nature of the lensing rate(missing out on lensing events does not matterso much if there are so many long ones that willstill be detectable).

Figure 8 depicts updated TMT lensing rates.A similar lack of time dependence is seen forTMT. Once again, extending out to at leasttens of parsecs allows for a substantial lensingrate, about 1 per 4 years with a conservativelimit of 23 magnitude; about 1 per year witha more liberal limit of 24 magnitude. Regard-less, certainly an improvement on the originalresults of Alexander and Sternberg.

6. FUTURE PLANS

Moving forward, there would be several waysto improve upon this calculation. I mentionedearlier that an average 3-D stellar velocity wasused in the updated calculation, adapted fromDo et al. (2013). As a future move, it wouldbe more correct to include a dependency in thevelocity distribution on distance from the GC,as a means to more accurately represent starsfound beyond the inner GC. Another meansfor progress would be to account more adeptlyfor factors which would prevent us from see-ing stars we should otherwise be able to detect(which are otherwise within the observationallimits of the telescope). This would includeadding in updated parameters for gas extinc-tion along with a mechanism of taking into ac-count source confusion. We cannot detect allthe stars predicted to exist, and the issue ofdetecting a source star against the confusion ofmultiple background stars (especially becausethe region of space being observed is so far andcrowded) is an important one to consider whencomputing an accurate rate. Finally, anotherstep to start considering is how we are goingto detect and register these lensing events, es-pecially if, as our rates have shown, multiple

events might be occurring over an observationalperiod of several years. We need to consider inparticular the time duration of lensing eventsand whether some might be overlapping or hap-pening at once. For microlensing, we needto consider how to differentiate these lensingevents from each other and from the regularbrightness fluctuations of the black hole. Forresolved lensing, we need to consider how weare going to detect the lensed pairs, especiallyconsidering source confusion. As NGAO andthe TMT become realities, these are questionswe need to consider in order to make microlens-ing and lensing detection a valid observationalendeavor.

7. CONCLUSIONS AND SIGNIFICANCE

We have thus discovered that detection of mi-crolensing and resolved lensing events from thesupermassive black hole at the Galactic cen-ter is a viable possibility for the future gener-ation of observational astronomy instruments:NGAO and TMT. Despite a rather low ratepredicted by Alexander and Sternberg in 1999,we sought to update and redo their calculationswith current parameters for the SMBH and theGC along with the detection limits of currentand future telescopes. At the moment, we canexpect to detect an event every few years withKeck AO and NGAO and perhaps about oneevent per year with the advent of TMT. Lens-ing events provide another means to observethe black hole, as we can only indirectly gaininformation on its nature by observing its inter-actions with objects around it. Observing thelensing by a black hole is yet another method ofconstraining parameters for its mass, distancefrom us, and the like. Bin-Nun (2010) suggeststhat the observing of microlensing events bythe SMBH is a plausible method for learningabout the black hole’s space-time metric andconstraining parameters for such a metric. Fi-nally, while we have good confirmation of theworkings of general relativity on a solar sys-tem and galactic scale (through other observa-tions of lensing done by stars and galaxies), wehave yet to confirm GR at the level of incred-ibly strong gravitational fields as found neara black hole. By observing a relativistic phe-nomenon done by the SMBH itself, we shouldbe able to use lensing as a useful probe of GRat the SMBH level: comparing the theoreticalpredictions of relativistic lensing against obser-

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 7

vational evidence. This is also an opportunityto discover where GR might break down andconsider alternate theories of gravity (includingquantum gravity). It appears that lensing andmicrolensing have the potential to take gravi-tational physics and GC astronomy into a newera of discovery of black hole and general rela-tivistic properties.

7.1. Acknowledgements

IL would like to thank foremost Dr. AndreaGhez and the UCLA Galactic Center Group fortheir mentorship and guidance. IL is gratefulto UCLA Physics and Astronomy as well andthe NSF for their offer of this research positionand granting of funding and resources. Specialthanks to Samantha Chappell, Breann Sitarski,Leo Meyer, and Gunther Witzel for their assis-tance in this project.

REFERENCES

2014, W. M. Keck Observatorys Next GenerationAdaptive Optics Facility. March 12, 2014, MSIP2014 Proposal

Alexander, T. & Sternberg, A. 1999, ApJ, 520, 137Bin-Nun, A. Y. 2010, Phys. Rev. D, 82, 064009Do, T., Martinez, G. D., Yelda, S., Ghez, A., Bullock,

J., Kaplinghat, M., Lu, J. R., Peter, A. H. G., &Phifer, K. 2013, ApJ, 779, L6

Ghez, A. M., Salim, S., Weinberg, N. N., Lu, J. R.,Do, T., Dunn, J. K., Matthews, K., Morris, M. R.,Yelda, S., Becklin, E. E., Kremenek, T.,Milosavljevic, M., & Naiman, J. 2008, ApJ, 689,1044

Schodel, R., Merritt, D., & Eckart, A. 2009, A&A,502, 91

Yelda, S., Meyer, L., Ghez, A., & Do, T. 2013, inProceedings of the Third AO4ELT Conference, ed.S. Esposito & L. Fini

8. FIGURES

8 Lipartito et al

Fig. 1.— The KLF generated by Alexander and Stern-berg. An expected K magnitude, 〈K〉 of 20.2 was foundby normalizing this distribution and finding the averagemagnitude.

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 9

Fig. 2.— Time-dependent microlensing rates for dif-ferent threshold magnitudes (ranging from 16 to 19),generated from reproducing the calculations of Alexan-der and Sternberg. Smaller dots are rates I producedand larger dots are the rates from Alexander and Stern-berg. Agreement is typically within an order of mag-nitude and is fairly consistent. Main result of 1-3 microlensing events per year in the limit of near-continuous observation is reproduced.

10 Lipartito et al

Fig. 3.— Time-dependent resolved lensing rates fordifferent threshold magnitudes (ranging from 16 to 19),generated from reproducing the calculations of Alexan-der and Sternberg. Smaller dots are rates I producedand larger dots are the rates from Alexander. Agree-ment is typically within an order of magnitude and isfairly consistent. Resolved lensing rates are an order ofmagnitude smaller than microlensing rates for all mag-nitude thresholds.

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 11

Fig. 4.— Time-dependent total rates (microlensingand resolved lensing) for different threshold magnitudes(ranging from 16 to 19), generated from reproducingthe calculations of Alexander and Sternberg. Smallerdots are rates I produced and larger dots are the ratesfrom Alexander and Sternberg. Agreement is typicallywithin an order of magnitude and is fairly consistent.Microlensing events dominate the total lensing rate ascan be seen by comparing the different rates.

12 Lipartito et al

Fig. 5.— The KLF for AO era Keck (blue) and NGAOera Keck (red). An expected K AO magnitude, 〈K〉 of18.5 was found by normalizing the AO distribution andfinding the average magnitude. An expected K NGAOmagnitude, 〈K〉 of 19.7 was found by normalizing theNGAO distribution and finding the average magnitude.

Calculating a Stellar-Supermassive Black Hole Microlensing Rate 13

Fig. 6.— The expected KLF for TMT era observa-tions. An expected K magnitude, 〈K〉 of 21.35 wasfound by normalizing this distribution and finding theaverage magnitude.

14 Lipartito et al

Fig. 7.— Time-dependent total lensing rates for dif-ferent Keck eras, up to 50 pc. These were generatedfrom updating the calculations of Alexander and Stern-berg. Lensing rates go up several orders of magnitudeas we move into Keck AO and NGAO eras

Fig. 8.— Time-dependent lensing rates (includ-ing both separate microlensing and resolved lensingrates) for different TMT magnitude limit estimates.These were generated from updating the calculationsof Alexander and Sternberg. Lensing rates for bothmagnitude limits are optimal if we can observe extend-ing out to include stars at least 10 pc from the GC.With the upper magnitude limit of 24, we can expectto nearly see 1 event per year.